Toniolo and Linder, Equation (10-)

Average Error: 47.6 → 5.0

Time: 22.8s

Precision: binary64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\ell \cdot 2}{\frac{k}{\cos k}}\\ t_3 := \left(\ell \cdot \frac{t_2}{k \cdot t_1}\right) \cdot \frac{1}{t}\\ \mathbf{if}\;k \leq -1 \cdot 10^{+170}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{k}}{k}}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{{t}^{-1}}}\\ \mathbf{elif}\;k \leq 3.570485739764951 \cdot 10^{+189}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{\frac{t_1}{\ell}} \cdot \frac{\frac{1}{k}}{t}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0))
        (t_2 (/ (* l 2.0) (/ k (cos k))))
        (t_3 (* (* l (/ t_2 (* k t_1))) (/ 1.0 t))))
   (if (<= k -1e+170)
     t_3
     (if (<= k 1e+60)
       (/
        (/ (* (* l 2.0) (/ (cos k) k)) k)
        (* (/ (sin k) l) (/ (sin k) (pow t -1.0))))
       (if (<= k 3.570485739764951e+189)
         t_3
         (* (/ t_2 (/ t_1 l)) (/ (/ 1.0 k) t)))))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

↓

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double t_2 = (l * 2.0) / (k / cos(k));
	double t_3 = (l * (t_2 / (k * t_1))) * (1.0 / t);
	double tmp;
	if (k <= -1e+170) {
		tmp = t_3;
	} else if (k <= 1e+60) {
		tmp = (((l * 2.0) * (cos(k) / k)) / k) / ((sin(k) / l) * (sin(k) / pow(t, -1.0)));
	} else if (k <= 3.570485739764951e+189) {
		tmp = t_3;
	} else {
		tmp = (t_2 / (t_1 / l)) * ((1.0 / k) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = (l * 2.0d0) / (k / cos(k))
    t_3 = (l * (t_2 / (k * t_1))) * (1.0d0 / t)
    if (k <= (-1d+170)) then
        tmp = t_3
    else if (k <= 1d+60) then
        tmp = (((l * 2.0d0) * (cos(k) / k)) / k) / ((sin(k) / l) * (sin(k) / (t ** (-1.0d0))))
    else if (k <= 3.570485739764951d+189) then
        tmp = t_3
    else
        tmp = (t_2 / (t_1 / l)) * ((1.0d0 / k) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double t_2 = (l * 2.0) / (k / Math.cos(k));
	double t_3 = (l * (t_2 / (k * t_1))) * (1.0 / t);
	double tmp;
	if (k <= -1e+170) {
		tmp = t_3;
	} else if (k <= 1e+60) {
		tmp = (((l * 2.0) * (Math.cos(k) / k)) / k) / ((Math.sin(k) / l) * (Math.sin(k) / Math.pow(t, -1.0)));
	} else if (k <= 3.570485739764951e+189) {
		tmp = t_3;
	} else {
		tmp = (t_2 / (t_1 / l)) * ((1.0 / k) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

↓

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	t_2 = (l * 2.0) / (k / math.cos(k))
	t_3 = (l * (t_2 / (k * t_1))) * (1.0 / t)
	tmp = 0
	if k <= -1e+170:
		tmp = t_3
	elif k <= 1e+60:
		tmp = (((l * 2.0) * (math.cos(k) / k)) / k) / ((math.sin(k) / l) * (math.sin(k) / math.pow(t, -1.0)))
	elif k <= 3.570485739764951e+189:
		tmp = t_3
	else:
		tmp = (t_2 / (t_1 / l)) * ((1.0 / k) / t)
	return tmp

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

↓

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(Float64(l * 2.0) / Float64(k / cos(k)))
	t_3 = Float64(Float64(l * Float64(t_2 / Float64(k * t_1))) * Float64(1.0 / t))
	tmp = 0.0
	if (k <= -1e+170)
		tmp = t_3;
	elseif (k <= 1e+60)
		tmp = Float64(Float64(Float64(Float64(l * 2.0) * Float64(cos(k) / k)) / k) / Float64(Float64(sin(k) / l) * Float64(sin(k) / (t ^ -1.0))));
	elseif (k <= 3.570485739764951e+189)
		tmp = t_3;
	else
		tmp = Float64(Float64(t_2 / Float64(t_1 / l)) * Float64(Float64(1.0 / k) / t));
	end
	return tmp
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

↓

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	t_2 = (l * 2.0) / (k / cos(k));
	t_3 = (l * (t_2 / (k * t_1))) * (1.0 / t);
	tmp = 0.0;
	if (k <= -1e+170)
		tmp = t_3;
	elseif (k <= 1e+60)
		tmp = (((l * 2.0) * (cos(k) / k)) / k) / ((sin(k) / l) * (sin(k) / (t ^ -1.0)));
	elseif (k <= 3.570485739764951e+189)
		tmp = t_3;
	else
		tmp = (t_2 / (t_1 / l)) * ((1.0 / k) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * 2.0), $MachinePrecision] / N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * N[(t$95$2 / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1e+170], t$95$3, If[LessEqual[k, 1e+60], N[(N[(N[(N[(l * 2.0), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.570485739764951e+189], t$95$3, N[(N[(t$95$2 / N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.00000000000000003e170 or 9.9999999999999995e59 < k < 3.57048573976495117e189

   1. Initial program 42.2

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Simplified 34.6

      \[\leadsto \color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(\ell \cdot \ell\right)} \]

   3. Taylor expanded in t around 0 21.4

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   4. Simplified 21.6

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}} \]

   5. Applied egg-rr 17.1

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(2 \cdot \frac{\cos k}{k \cdot k}\right)}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}} \]

   6. Applied egg-rr 10.3

      \[\leadsto \frac{\color{blue}{\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{k}}{k}}}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}} \]

   7. Applied egg-rr 7.8

      \[\leadsto \color{blue}{\left(\frac{\frac{\ell \cdot 2}{\frac{k}{\cos k}}}{{\sin k}^{2} \cdot k} \cdot \ell\right) \cdot \frac{1}{t}} \]

   if -1.00000000000000003e170 < k < 9.9999999999999995e59

   1. Initial program 55.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Simplified 46.1

      \[\leadsto \color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(\ell \cdot \ell\right)} \]

   3. Taylor expanded in t around 0 23.6

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   4. Simplified 22.1

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}} \]

   5. Applied egg-rr 8.6

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(2 \cdot \frac{\cos k}{k \cdot k}\right)}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}} \]

   6. Applied egg-rr 7.7

      \[\leadsto \frac{\color{blue}{\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{k}}{k}}}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}} \]

   7. Applied egg-rr 2.7

      \[\leadsto \frac{\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{k}}{k}}{\color{blue}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{{t}^{-1}}}} \]

   if 3.57048573976495117e189 < k

   1. Initial program 37.2

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Simplified 32.7

      \[\leadsto \color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(\ell \cdot \ell\right)} \]

   3. Taylor expanded in t around 0 22.1

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   4. Simplified 22.2

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}} \]

   5. Applied egg-rr 20.4

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(2 \cdot \frac{\cos k}{k \cdot k}\right)}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}} \]

   6. Applied egg-rr 13.4

      \[\leadsto \frac{\color{blue}{\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{k}}{k}}}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}} \]

   7. Applied egg-rr 6.0

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot 2}{\frac{k}{\cos k}}}{\frac{{\sin k}^{2}}{\ell}} \cdot \frac{\frac{1}{k}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{+170}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{\ell \cdot 2}{\frac{k}{\cos k}}}{k \cdot {\sin k}^{2}}\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;k \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{k}}{k}}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{{t}^{-1}}}\\ \mathbf{elif}\;k \leq 3.570485739764951 \cdot 10^{+189}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{\ell \cdot 2}{\frac{k}{\cos k}}}{k \cdot {\sin k}^{2}}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{\frac{k}{\cos k}}}{\frac{{\sin k}^{2}}{\ell}} \cdot \frac{\frac{1}{k}}{t}\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))